How many different counting numbers will each leave a remainder of 5 when divided into 47?
Solution: For a number to leave a remainder of 5 when divided into 47, it must satisfy two conditions:

1. it must divide exactly into $47 - 5$, or 42, and

2. it must be greater than 5, because the divisor is always greater than the remainder.

We list all divisors of 42 in pairs. They are 1 and 42, 2 and 21, 3 and 14, 6 and 7. Of these, only 42, 21, 14, 6, and 7 are greater than 5. There are $\boxed{5}$ different counting numbers that will leave a remainder of 5 when divided into 47.